New method to calculate the n-particle irreducible effective action

In this paper, we present a new method to calculate the n-Loop n-particle irreducible effective action. The key is an organizational trick that involves the introduction of a set of fictitious bare vertices that are set to zero at the end of the calculation. Using these fictitious vertices, we prove that the Schwinger-Dyson equations are the same as the equations of motion obtained from the n-particle irreducible effective action, up to the level at which they respect the symmetries of the original theory. This result allows us to obtain the effective action directly from the Schwinger-Dyson equations, which are comparatively easy to calculate. As a check of our method, we reproduce the known results for the n-Loop n-particle irreducible effective action with n = 4 and n = 5. We also use the technique to calculate the 6-Loop 6-particle irreducible effective action.